Subgroups — C₁₃×#2413 #13111471

Order 1664.

610 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#123	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#123	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#398	C₂	1	2	yes	maximal, complemented
6	C₁₃×#396	C₂	1	2	yes	maximal, complemented
7	C₁₃×#389	C₂	1	2	yes	maximal, complemented
8	C₁₃×#388	C₂	1	2	yes	maximal, complemented
9	C₁₃×#386	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#385	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#123	—	2	4	no	—
17	C₁₃×#123	—	2	4	no	—
18	C₁₃×#117	—	2	4	no	—
19	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
32	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
35	C₁₃×#95	C₂²	1	2	yes	—
36	C₁₃×#95	C₂²	1	2	yes	—
37	C₁₃×#95	C₂²	1	2	yes	—
38	C₁₃×#95	C₂²	1	2	yes	—
39	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	—	2	2	no	—
43	C₂×D₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	—	2	2	no	—
45	C₂²×C₄×C₁₃	—	2	2	no	—
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	D₄	1	4	yes	—
52	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
61	C₁₃×C₄⋊C₄	—	2	4	no	—
62	C₁₃×C₄⋊C₄	—	2	4	no	—
63	C₁₃×C₄⋊C₄	—	2	4	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	2	4	no	—
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	2	no	—
71	C₁₃×C₂²⋊C₄	—	2	2	no	—
72	C₁₃×C₂²⋊C₄	—	2	4	no	—
73	C₁₃×C₂²⋊C₄	—	2	4	no	—
74	C₁₃×C₂²⋊C₄	D₄	1	4	yes	—
75	C₄²×C₁₃	—	2	4	no	—
76	#2413	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₁₃	—	2	2	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	—	2	2	no	—
84	C₂³×C₁₃	C₂×D₄	1	2	yes	—
85	C₂³×C₁₃	C₂×D₄	1	2	yes	—
86	C₂³×C₁₃	#41	1	2	yes	—
87	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
88	D₄×C₁₃	—	4	4	no	—
89	C₂×C₄×C₁₃	—	2	8	no	—
90	C₂×C₄×C₁₃	—	2	8	no	—
91	C₂×C₄×C₁₃	—	2	1	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
114	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	#41	1	4	yes	—
117	C₂×#123	C₂×C₁₃	1	1	yes	characteristic
118	C₂×#123	C₂×C₁₃	1	1	yes	characteristic
119	C₂×#117	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂²×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#398	C₂×C₁₃	1	2	yes	complemented
122	#396	C₂×C₁₃	1	2	yes	complemented
123	#389	C₂×C₁₃	1	2	yes	complemented
124	#388	C₂×C₁₃	1	2	yes	complemented
125	#386	C₂×C₁₃	1	1	yes	characteristic
126	#385	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#381	C₂×C₁₃	1	1	yes	characteristic, complemented
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	2	4	no	—
132	C₂²×C₁₃	—	4	1	no	—
133	C₂²×C₁₃	—	4	1	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	—	4	2	no	—
137	C₂²×C₁₃	#121	1	4	yes	—
138	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×#41	1	2	yes	—
142	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
143	C₄×C₁₃	—	2	2	no	—
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	4	2	no	—
148	C₄×C₁₃	—	4	1	no	—
149	C₄×C₁₃	—	4	1	no	—
150	C₄×C₁₃	—	4	1	no	—
151	C₄×C₁₃	—	4	2	no	—
152	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
153	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
154	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
156	#123	—	2	4	no	—
157	#123	—	2	4	no	—
158	#117	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
162	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
163	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
175	#95	C₂²×C₁₃	1	2	yes	—
176	#95	C₂²×C₁₃	1	2	yes	—
177	#95	C₂²×C₁₃	1	2	yes	—
178	#95	C₂²×C₁₃	1	2	yes	—
179	C₂×C₁₃	—	2	4	no	—
180	C₂×C₁₃	—	8	1	no	—
181	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
182	C₂×C₁₃	#525	1	1	yes	characteristic
183	C₂×C₁₃	#536	1	1	yes	characteristic
184	C₂×C₁₃	#547	1	2	yes	—
185	C₂×C₁₃	#552	1	2	yes	—
186	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
187	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	D₄×C₁₃	1	4	yes	—
199	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
201	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
203	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
205	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
207	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	4	no	—
221	C₂²⋊C₄	D₄×C₁₃	1	4	yes	—
222	C₄²	—	2	4	no	—
223	C₁₃	#2413	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×D₄×C₁₃	1	2	yes	—
232	C₂³	C₂×D₄×C₁₃	1	2	yes	—
233	C₂³	C₁₃×#41	1	2	yes	—
234	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
235	D₄	—	4	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
261	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
262	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
263	C₂×C₄	C₁₃×#41	1	4	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	C₁₃×#121	1	4	yes	—
274	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
275	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
276	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
277	C₂²	C₂×C₁₃×#41	1	2	yes	—
278	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	4	no	—
283	C₄	—	4	2	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	4	1	no	—
287	C₄	—	4	2	no	—
288	C₂	—	2	4	no	—
289	C₂	—	8	1	no	—
290	C₂	C₂×C₁₃×#121	1	1	yes	characteristic
291	C₂	C₁₃×#525	1	1	yes	characteristic
292	C₂	C₁₃×#536	1	1	yes	characteristic
293	C₂	C₁₃×#547	1	2	yes	—
294	C₂	C₁₃×#552	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂²×D₄×C₁₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₂×C₁₃×#117	C₂	semidirect
4. C₂²×C₁₃×C₄⋊C₄	C₂	semidirect
5. C₁₃×#398	C₂	semidirect
6. C₁₃×#396	C₂	semidirect
7. C₁₃×#389	C₂	semidirect
8. C₁₃×#388	C₂	semidirect
10. C₁₃×#385	C₂	semidirect
11. C₁₃×#381	C₂	semidirect
76. #2413	C₁₃	direct (N × H)
119. C₂×#117	C₂×C₁₃	semidirect
120. C₂²×C₄⋊C₄	C₂×C₁₃	semidirect
121. #398	C₂×C₁₃	semidirect
122. #396	C₂×C₁₃	semidirect
123. #389	C₂×C₁₃	semidirect
124. #388	C₂×C₁₃	semidirect
126. #385	C₂×C₁₃	semidirect
127. #381	C₂×C₁₃	semidirect
223. C₁₃	#2413	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#123	2
C₂×C₁₃×#117	2
C₂²×C₁₃×C₄⋊C₄	2
C₁₃×#398	2
C₁₃×#396	2
C₁₃×#389	2
C₁₃×#388	2
C₁₃×#386	2
C₁₃×#385	2
C₁₃×#381	2
#2413	13

Subgroups — C13×#2413 #13111471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2413 #13111471